Factor D Decimation MCQ [Free PDF] – Objective Question Answer for Factor D Decimation Quiz

We know that if we reduce the sampling rate simply by selecting every Dth value of x(n), the signal will be an aliased version of x(n).

We know that if we reduce the sampling rate simply by selecting every Dth value of x(n), the signal will be an aliased version of x(n), with a folding frequency of F/2D.

To avoid aliasing, we must reduce the bandwidth of x(n) to Fmax=F/2D. Then we may down-sample by D and thus avoid sampling.

Although the filtering operation on x(n) is linear and time-invariant, the down-sampling operation in combination with the filtering results in a time-variant system.

Given the fact that x(n) produces y(m), we note that x(n-n0) does not imply y(n-n0) unless n0 is a multiple of D.

Given the fact that x(n) produces y(m), we note that x(n-n0) does not imply y(n-n0) unless n0 is a multiple of D. Consequently, the overall linear operation, that is linear filtering followed by downsampling on x(n) is not time-invariant.

We know that the equation for y(m) is given as
y(m)= (bar{v}(mD.)= v(mD..p(mD..

The frequency-domain characteristic of the output sequence y(m) can be obtained by relating the spectrum of y(m) to the spectrum of the input sequence x(n).

(bar{v}(n)) can be viewed as a sequence obtained by multiplying v(n) with a periodic train of impulses p(n) with period D.

The input sequence x(n) is passed through a low pass filter, characterized by the impulse response h(n) and a frequency response HD(ω), which ideally satisfies the condition,
HD(ω)=1, |ω| ≤ π/D
=0, otherwise.

The input sequence x(n) is passed through a low pass filter, characterized by the impulse response h(n) and a frequency response HD(ω), which ideally satisfies the condition,

HD(ω)=1, |ω| ≤ π/D
=0, otherwise
Thus the filter eliminates the spectrum of X(ω) in the range π/D < ω < π.

In telecommunication systems that transmit and receive different types of signals, there is a requirement to process the various signals at different rates commensurate with the corresponding bandwidths of the signals.

The process of converting a signal from a given rate to a different rate is known as sampling rate conversion.

Systems that employ multiple sampling rates in the processing of digital signals are called multi-rate digital signal processing systems.

Sampling rate conversion of a digital signal can be accomplished in one of the two general methods. One method is to pass the signal through the D/A converter, filter it if necessary, and then resample the resulting analog signal at the desired rate. The second method is to perform the sampling rate conversion entirely in the digital domain.

One apparent advantage of the given method is that the new sampling rate can be arbitrarily selected and need not have any special relationship with the old sampling rate.

The major disadvantage of the given type of conversion is the signal distortion introduced by the D/A converter in the signal reconstruction and by the quantization effects in the A/D conversion.

There are several applications of sampling rate conversion in multi-rate digital signal processing systems, which include the implementation of narrowband filters, quadrature mirror filters, and digital filter banks.

There are many applications where quadrature mirror filters can be used. Some of these applications are sub-band coding, trans-multiplexers, and many other applications.

The process of sampling rate conversion in the digital domain can be viewed as a linear filtering operation as illustrated in the given figure.

The input signal x(n) is characterized by the sampling rate Fx and the output signal y(m) is characterized by the sampling rate Fy, then
Fy/Fx = I/D

where I and D are relatively prime integers.

The process of reducing the sampling rate by a factor D, i.e., down-sampling by D is called as decimation.

The process of increasing the sampling rate by an integer factor I, i.e., up-sampling by I is called as interpolation.